Towards realistic simulations of lava dome growth using the level set method

The level set method has been implemented in a computational volcanology context. New techniques are presented to solve the advection equation and the reinitialisation equation. These techniques are based upon an algorithm developed in the finite difference context, but are modified to take advantage of the robustness of the finite element method. The resulting algorithm is tested on a well documented Rayleigh–Taylor instability benchmark [19], and on an axisymmetric problem where the analytical solution is known. Finally, the algorithm is applied to a basic study of lava dome growth.

Using the level set method to model endogenous lava dome growth

Modeling volcanic phenomena is complicated by free-surfaces often supporting large rheological gradients. Analytical solutions and analogue models provide explanations for fundamental characteristics of lava flows. But more sophisticated models are needed, incorporating improved physics and rheology to capture realistic events. To advance our understanding of the flow dynamics of highly viscous lava in Peléean lava dome formation, axi-symmetrical Finite Element Method (FEM) models of generic endogenous dome growth have been developed. We use a novel technique, the level-set method, which tracks a moving interface, leaving the mesh unaltered. The model equations are formulated in an Eulerian framework. In this paper we test the quality of this technique in our numerical scheme by considering existing analytical and experimental models of lava dome growth which assume a constant Newtonian viscosity. We then compare our model against analytical solutions for real lava domes extruded on Soufrière, St. Vincent, W.I. in 1979 and Mount St. Helens, USA in October 1980 using an effective viscosity. The level-set method is found to be computationally light and robust enough to model the free-surface of a growing lava dome. Also, by modeling the extruded lava with a constant pressure head this naturally results in a drop in extrusion rate with increasing dome height, which can explain lava dome growth observables more appropriately than when using a fixed extrusion rate. From the modeling point of view, the level-set method will ultimately provide an opportunity to capture more of the physics while benefiting from the numerical robustness of regular grids.

Growth of the lava dome and extrusion rates at Soufrière Hills Volcano, Montserrat, West Indies: 2005–2008

The third episode of lava dome growth at Soufrière Hills Volcano began 1 August 2005 and ended 20 April 2007. Volumes of the dome and talus produced were measured using a photo-based method with a calibrated camera for increased accuracy. The total dense rock equivalent (DRE) volume of extruded andesite magma (306 ± 51 Mm3) was similar within error to that produced in the earlier episodes but the average extrusion rate was 5.6 ± 0.9 m3s−1 (DRE), higher than the previous episodes. Extrusion rates varied in a pulsatory manner from <0.5 m3s−1 to ∼20 m3s−1. On 18 May 2006, the lava dome had reached a volume of 85 Mm3 DRE and it was removed in its entirety during a massive dome collapse on 20 May 2006. Extrusion began again almost immediately and built a dome of 170 Mm3 DRE with a summit height 1047 m above sea level by 4 April 2007. There were few moderate-sized dome collapses (1–10 Mm3) during this extrusive episode in contrast to the first episode of dome growth in 1995–8 when they were numerous. The first and third episodes of dome growth showed a similar pattern of low (<0.5 m3s−1) but increasing magma flux during the early stages, with steady high flux after extrusion of ∼25 Mm3

Modelling the lava dome extruded at Soufrière Hills Volcano, Montserrat, August 2005–May 2006. Part I: Dome shape and internal structure

Lava domes comprise core, carapace, and clastic talus components. They can grow endogenously by inflation of a core and/or exogenously with the extrusion of shear bounded lobes and whaleback lobes at the surface. Internal structure is paramount in determining the extent to which lava dome growth evolves stably, or conversely the propensity for collapse. The more core lava that exists within a dome, in both relative and absolute terms, the more explosive energy is available, both for large pyroclastic flows following collapse and in particular for lateral blast events following very rapid removal of lateral support to the dome. Knowledge of the location of the core lava within the dome is also relevant for hazard assessment purposes. A spreading toe, or lobe of core lava, over a talus substrate may be both relatively unstable and likely to accelerate to more violent activity during the early phases of a retrogressive collapse. Soufrière Hills Volcano, Montserrat has been erupting since 1995 and has produced numerous lava domes that have undergone repeated collapse events. We consider one continuous dome growth period, from August 2005 to May 2006 that resulted in a dome collapse event on 20th May 2006. The collapse event lasted 3 h, removing the whole dome plus dome remnants from a previous growth period in an unusually violent and rapid collapse event. We use an axisymmetrical computational Finite Element Method model for the growth and evolution of a lava dome. Our model comprises evolving core, carapace and talus components based on axisymmetrical endogenous dome growth, which permits us to model the interface between talus and core. Despite explicitly only modelling axisymmetrical endogenous dome growth our core–talus model simulates many of the observed growth characteristics of the 2005–2006 SHV lava dome well. Further, it is possible for our simulations to replicate large-scale exogenous characteristics when a considerable volume of talus has accumulated around the lower flanks of the dome. Model results suggest that dome core can override talus within a growing dome, potentially generating a region of significant weakness and a potential locus for collapse initiation.

Modelling the lava dome extruded at Soufrière Hills Volcano, Montserrat, August 2005–May 2006. Part II: Rockfall activity and talus deformation

During many lava dome-forming eruptions, persistent rockfalls and the concurrent development of a substantial talus apron around the foot of the dome are important aspects of the observed activity. An improved understanding of internal dome structure, including the shape and internal boundaries of the talus apron, is critical for determining when a lava dome is poised for a major collapse and how this collapse might ensue. We consider a period of lava dome growth at the Soufrière Hills Volcano, Montserrat, from August 2005 to May 2006, during which a 100 × 106 m3 lava dome developed that culminated in a major dome-collapse event on 20 May 2006. We use an axi-symmetrical Finite Element Method model to simulate the growth and evolution of the lava dome, including the development of the talus apron. We first test the generic behaviour of this continuum model, which has core lava and carapace/talus components. Our model describes the generation rate of talus, including its spatial and temporal variation, as well as its post-generation deformation, which is important for an improved understanding of the internal configuration and structure of the dome. We then use our model to simulate the 2005 to 2006 Soufrière Hills dome growth using measured dome volumes and extrusion rates to drive the model and generate the evolving configuration of the dome core and carapace/talus domains. The evolution of the model is compared with the observed rockfall seismicity using event counts and seismic energy parameters, which are used here as a measure of rockfall intensity and hence a first-order proxy for volumes. The range of model-derived volume increments of talus aggraded to the talus slope per recorded rockfall event, approximately 3 × 103–13 × 103 m3 per rockfall, is high with respect to estimates based on observed events. From this, it is inferred that some of the volumetric growth of the talus apron (perhaps up to 60–70%) might have occurred in the form of aseismic deformation of the talus, forced by an internal, laterally spreading core. Talus apron growth by this mechanism has not previously been identified, and this suggests that the core, hosting hot gas-rich lava, could have a greater lateral extent than previously considered.

Clastic and core lava components of a silicic lava dome

The formation of a lava dome involves fractionation of the lava into core and clastic components. We show that for three separate, successive andesitic lava domes that grew at Soufrière Hills volcano, Montserrat, between 1999 and 2007, the volumetric proportion of the lava converted to talus or pyroclastic flow deposits was 50%–90% of the lava extruded. Currently, only 8% of the total magma extruded during the 1995–2007 eruption remains as core lava. The equivalent representation in the geological record will probably be even lower. Most of the lava extruded at the surface flowed no further than 150–300 m from the vent before disaggregation, resulting in a lava core whose shape tends to a cylinder. Moderate to high extrusion rates at the Soufrière Hills domes may have contributed to the large clastic fraction observed. Creating talus dissipates much of the energy that would otherwise be stored in the core lava of domes. The extreme hazards from large pyroclastic flows and blasts posed by wholesale collapse of a lava dome depend largely on the size of the lava core, and hence on the aggregate history of the partitioning process, not on the size of the dome.

Cyclic extrusion of a lava dome based on a stick-slip mechanism

Lava dome eruptions are sometimes characterised by large periodic fluctuations in extrusion rate over periods of hours that may be accompanied by Vulcanian explosions and pyroclastic flows. We consider a simple system of nonlinear equations describing a 1D flow of lava extrusion through a deep elastic dyke feeding a shallower cylindrical conduit in order to simulate this short-period cyclicity. Stick-slip conditions depending on a critical shear stress are assumed at the wall boundary of the cylindrical conduit. By analogy with the behaviour of industrial polymers in a plastic extruder, the elastic dyke acts like a barrel and the shallower cylindrical portion of the conduit as a die for the flow of magma acting as a polymer. When we applied the model to the Soufrière Hills Volcano, Montserrat, for which the key parameters have been evaluated from previous studies, cyclic extrusions with periods from 3 to 30 h were readily simulated, matching observations. The model also reproduces the reduced period of cycles observed when a major unloading event occurs due to lava dome collapse.

Coupled subdaily and multiweek cycles during the lava dome erupt of Soufriere Hills Volcano, Montserrat

Observations of volcanoes extruding andesitic lava to produce lava domes often reveal cyclic behaviour. At Soufriere Hills Volcano, Montserrat, cycles with sub-daily and multi-week periods have been recognised on many occasions. These two types of cycle have been modelled separately as stick-slip magma flow at the junction between a dyke and an overlying cylindrical conduit (Costa et al. 2012), and as the filling and discharge of magma through the elastic-walled dyke (Costa et al., 2007a) respectively. Here, we couple these two models to simulate the behaviour over a period of well-observed multi-week cycles, with accompanying sub-daily cycles, from 13 May to 21 September 1997. The coupled model captures well the asymmetrical first-order behaviour: the first 40% of the multi-week cycle consists of high rates of lava extrusion during short period/high amplitude sub-daily cycles as the dyke reservoir discharges itself. The remainder of the cycle involves increasing pressurization as more magma is stored, and extrusion rate falls, followed by a gradual increase in the period of the sub-daily cycles.

Rapid topographic change measured by high-resolution satellite radar at Soufriere Hills Volcano, Montserrat, 2008-2010

High-resolution satellite radar observations of erupting volcanoes can yield valuable information on rapidly changing deposits and geomorphology. Using the TerraSAR-X (TSX) radar with a spatial resolution of about 2 m and a repeat interval of 11-days, we show how a variety of techniques were used to record some of the eruptive history of the Soufriere Hills Volcano, Montserrat between July 2008 and February 2010. After a 15-month pause in lava dome growth, a vulcanian explosion occurred on 28 July 2008 whose vent was hidden by dense cloud. We were able to show the civil authorities using TSX change difference images that this explosion had not disrupted the dome sufficient to warrant continued evacuation. Change difference images also proved to be valuable in mapping new pyroclastic flow deposits: the valley-occupying block-and-ash component tending to increase backscatter and the marginal surge deposits reducing it, with the pattern reversing after the event. By comparing east- and west-looking images acquired 12 hours apart, the deposition of some individual pyroclastic flows can be inferred from change differences. Some of the narrow upper sections of valleys draining the volcano received many tens of metres of rockfall and pyroclastic flow deposits over periods of a few weeks. By measuring the changing shadows cast by these valleys in TSX images the changing depth of infill by deposits could be estimated. In addition to using the amplitude data from the radar images we also used their phase information within the InSAR technique to calculate the topography during a period of no surface activity. This enabled areas of transient topography, crucial for directing future flows, to be captured.

Sensitivity of OMI SO2 measurements to variable eruptive behaviour at Soufrière Hills Volcano, Montserrat

Since 2004, the satellite-borne Ozone Mapping Instrument (OMI) has observed sulphur dioxide (SO2) plumes during both quiescence and effusive eruptive activity at Soufrière Hills Volcano, Montserrat. On average, OMI detected a SO2 plume 4-6 times more frequently during effusive periods than during quiescence in the 2008-2010 period. The increased ability of OMI to detect SO2 during eruptive periods is mainly due to an increase in plume altitude rather than a higher SO2 emission rate. Three styles of eruptive activity cause thermal lofting of gases (Vulcanian explosions; pyroclastic flows; a hot lava dome) and the resultant plume altitudes are estimated from observations and models. Most lofting plumes from Soufrière Hills are derived from hot domes and pyroclastic flows. Although Vulcanian explosions produced the largest plumes, some produced only negligible SO2 signals detected by OMI. OMI is most valuable for monitoring purposes at this volcano during periods of lava dome growth and during explosive activity.

The influence of viscous and latent heating on crystal-rich magma flow in a conduit

The flow dynamics of crystal-rich high-viscosity magma is likely to be strongly influenced by viscous and latent heat release. Viscous heating is observed to play an important role in the dynamics of fluids with temperature-dependent viscosities. The growth of microlite crystals and the accompanying release of latent heat should play a similar role in raising fluid temperatures. Earlier models of viscous heating in magmas have shown the potential for unstable (thermal runaway) flow as described by a Gruntfest number, using an Arrhenius temperature dependence for the viscosity, but have not considered crystal growth or latent heating. We present a theoretical model for magma flow in an axisymmetric conduit and consider both heating effects using Finite Element Method techniques. We consider a constant mass flux in a 1-D infinitesimal conduit segment with isothermal and adiabatic boundary conditions and Newtonian and non-Newtonian magma flow properties. We find that the growth of crystals acts to stabilize the flow field and make the magma less likely to experience a thermal runaway. The additional heating influences crystal growth and can counteract supercooling from degassing-induced crystallization and drive the residual melt composition back towards the liquidus temperature. We illustrate the models with results generated using parameters appropriate for the andesite lava dome-forming eruption at Soufriere Hills Volcano, Montserrat. These results emphasize the radial variability of the magma. Both viscous and latent heating effects are shown to be capable of playing a significant role in the eruption dynamics of Soufriere Hills Volcano. Latent heating is a factor in the top two kilometres of the conduit and may be responsible for relatively short-term (days) transients. Viscous heating is less restricted spatially, but because thermal runaway requires periods of hundreds of days to be achieved, the process is likely to be interrupted. Our models show that thermal evolution of the conduit walls could lead to an increase in the effective diameter of flow and an increase in flux at constant magma pressure.

Lava production at Soufriere Hills Volcano, Montserrat: 1995-2009

[1] We estimate that about 1 km3 of andesitic lava has been produced at Soufrière Hills Volcano, Montserrat from 1995 to 2009. There were three major episodes of extrusion, each lasting about 2 to 3.5 years and producing about 280 to 340 M m3 of lava, and one minor episode. Our estimates account for the dense rock equivalent volumetric contributions from the core and talus components of the lava dome, pyroclastic flow deposits and air-fall deposits. By 2005 at least two thirds of the erupted mass has already entered the sea. The average lava flux across the major extrusion episodes has been 3–5 m3s−1, with short-period (10–15 days) pulses up to 10–20 m3s−1. The first and third episodes of extrusion show similar flux histories suggesting similar behaviour of the system ten years apart. Waning flux towards the end of each episode may be caused by declining overpressure in the magma reservoir.